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Towards Positive Geometry of Multi Scalar Field Amplitudes : Accordiohedron and Effective Field Theory

The geometric structure of S-matrix encapsulated by the "Amplituhedron program" has begun to reveal itself even in non-supersymmetric quantum field theories. Starting with the seminal work of Arkani-Hamed, Bai, He and Yan it is now understood that for a wide class of scalar quantum field theories, tree-level amplitudes are canonical forms associated to polytopes known as accordiohedra. Similarly the higher loop scalar integrands are canonical forms associated to so called type-D cluster polytopes for cubic interactions or recently discovered class of polytopes termed pseudo-accordiohedron for higher order scalar interactions. In this paper, we continue to probe the universality of these structures for a wider class of scalar quantum field theories. More in detail, we discover new realisations of the associahedron in planar kinematic space whose canonical forms generate (colour-ordered) tree-level S matrix of external massless particles with $n-4$ massless poles and one massive pole at $m^{2}$. The resulting amplitudes are associated to $λ_{1}\, ϕ_{1}^{3}\, +\, λ_{2}\, ϕ_{1}^{2}ϕ_{2}$ potential where $ϕ_{1}$ and $ϕ_{2}$ are massless and massive scalar fields with bi-adjoint colour indices respectively. We also show how in the "decoupling limit" (where $m \rightarrow \infty, λ_{2} \rightarrow \infty$ such that $g := \frac{λ_{2}}{m} = \textrm{finite}$) these associahedra project onto a specific class of accordiohedron which are known to be positive geometries of amplitudes generated by $λϕ_{1}^{3} + g ϕ_{1}^{4}$.